Problem: Simplify; express your answer in exponential form. Assume $k\neq 0, z\neq 0$. $\dfrac{{(k^{5})^{-5}}}{{(kz^{3})^{-3}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${k^{5}}$ to the exponent ${-5}$ . Now ${5 \times -5 = -25}$ , so ${(k^{5})^{-5} = k^{-25}}$ In the denominator, we can use the distributive property of exponents. ${(kz^{3})^{-3} = (k)^{-3}(z^{3})^{-3}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(k^{5})^{-5}}}{{(kz^{3})^{-3}}} = \dfrac{{k^{-25}}}{{k^{-3}z^{-9}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{-25}}}{{k^{-3}z^{-9}}} = \dfrac{{k^{-25}}}{{k^{-3}}} \cdot \dfrac{{1}}{{z^{-9}}} = k^{{-25} - {(-3)}} \cdot z^{- {(-9)}} = k^{-22}z^{9}$.